函数作业代写 MAT401 Homework Problems

MAT401

函数作业代写 Show that Gal(E/Q) is a cyclic group of order 12 and determine a generator in the form τ = σa in notation analogous to the Q(ζ7)

Read: Gallian, Chapter 32 to p 536 and the Notes on Fields Extensions.

These problems are due on Crowdmark by 6pm on Wednesday, March 30th. The solutions will be discussed in tutorials that week.

Provide the details determining the fifixed fifield of the subgroup example in the notes.

函数作业代写

(i) Show that Gal(E/Q) is a cyclic group of order 12 and determine a generator in the form τ = σa in notation analogous to the Q(ζ7) example.

(ii) Write down the lattice of subgroups of Gal(E/Q).

(iii) Show that E contains a unique quadratic (i.e., degree 2) extension Q(√d) of Q and determine what it is, i.e., what is d?

Let f(x) ∈ Q[x] be an irreducible polynomial of degree n and let E be a splitting fifield of f(x) over Q. Suppose that α is root of f(x) in E, but that Q (α) 6=

E. Show that Gal( E//Q) is not an abelian group. 函数作业代写

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